I think it's an unspoken rule that an unelected head of state should never comment on political matters. In the UK Blair apologised for the Irish famine and Brown apologised for the Atlantic slave trade but the queen remained slient on both matters.

I'm on the fence as to if Japan needs to make further statements of regret/apology towards the Korean people whatever the wording. But if it happens it should come from the elected PM and not an outdated institution.

If the Emperor were such an outdated institution, why does he still reign at all?

I agree with Fox upthread. The Emperor connects the current Japanese polity to the past in a way Japan's disposable Prime Ministers cannot. Furthermore, a previous Emperor actually wielded considerable power and influence during WWII when the atrocities were committed.